For this post, A is alpha, B is beta, and Y is gamma, and represent the angles of the triangle. Little a, b, and c represent the sides opposite them, respectively. The form of the formula is:
Code:
a - b sin(1/2[A - B]
----- = -------------
c cos([1/2]Y)
There is a step in the derivation that I don't understand. Here is the derivation from my solutions book, and so this post doesn't get too protracted, I'll stop at the point that I don't understand:
a - b a b
----- = -- - --
c c c
= sin A sin B
---- - ----
sin Y sin Y
= sin A - sin B
-------------
sin Y
= 2sin( (A-B) / 2) cos( (A+B) / 2)
------------------------------ makes use of sum to product formula
sin (2Y/2)
= 2sin( (A-B) / 2) cos( (A+B) / 2)
--------------------------------
2sin(Y/2)cos(Y/2)
= sin( (A-B) / 2) cos(PI/2 - Y/2)
--------------------------------
sin(Y/2)cos(Y/2)
How does cos((A+B)/2) get turned into cos(PI/2 - Y/2)?